Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case

Abstract

We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order $i{u}_t+{\Delta }^{2}u=|u{|}^{\frac{8}{d-4}}u$. We prove that if a maximal-lifespan radial solution $u:I\times {\mathbb{R}}^d\to \mathbb{C}$ obeys $\underset{t\in I}{\sup }\Vert \Delta u(t){\Vert }_2<\Vert \Delta W{\Vert }_2$, then it is global and scatters both forward and backward in time. Here $W$ denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state $W$ at some point in time, then the solution is global and scatters.

37 pages, no figure

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• On defocusing fourth-order coupled nonlinear Schrodinger equations
• The Fourth-order dispersive nonlinear Schrödinger equation: orbital stability of a standing wave
• Scattering for the focusing $L^2$ -supercritical and ${\dot{H}}^2$-subcritical biharmonic NLS Equations
• Refined Scattering and Hermitian Spectral Theory for Linear Higher-Order Schrödinger Equations
• Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient
• Blowup of cylindrically symmetric solutions for biharmonic NLS
• Random data theory for the cubic fourth-order nonlinear Schrödinger equation
• On focusing coupled fourth-order nonlinear Schrodinger equations
• Global existence for the defocusing mass-critical nonlinear fourth-order Schrödinger equation below the energy space
• Global solutions for $H^s$-critical nonlinear biharmonic Schrödinger equation
• The mass-critical fourth-order Schrodinger equation in high dimensions